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Kick-then-drift symplectic Euler on the harmonic oscillator

Example

For $H (q, p) = \frac{1}{2} (q^{2} + p^{2})$ on $R^{2}$ and $h > 0$ , the kick-then-drift symplectic Euler step is $(\begin{array}{c} q^{+} \\ p^{+} \end{array}) = (\begin{array}{cc} 1 - h^{2} & h \\ - h & 1 \end{array}) (\begin{array}{c} q \\ p \end{array}) .$ The same matrix formula is its algebraic signed-step extension, and that extension is symplectic for every real $h$ . The original Hamiltonian need not be exactly conserved.

Facts & Assumptions

Given: $V (q) = q^{2} / 2$ , $M = 1$ , the local kick-then-drift algorithm, and $J = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ .

[F1]

The method applies $p^{+} = p - h V^{'} (q)$ and then $q^{+} = q + h p^{+}$ . Local kick-then-drift symplectic Euler step

[F2]

For $S = (\begin{array}{cc} a & b \\ c & d \end{array})$ , multiplication gives $S^{T} J S = (\begin{array}{cc} 0 & a d - b c \\ b c - a d & 0 \end{array}) = (\det S) J$ .

Verification

1.1givenF1algebra

Since $V^{'} (q) = q$ , [F1] gives $p^{+} = p - h q$ and $q^{+} = q + h (p - h q) = (1 - h^{2}) q + h p$ , which is the displayed matrix.

2.1F2step 1.1algebra

Its determinant is $(1 - h^{2}) \cdot 1 - h (- h) = 1$ . By [F2], $S^{T} J S = J$ , verifying symplecticity.

3.1step 1.1givenalgebra∎

For any $h > 0$ , starting from $(q, p) = (0, 1)$ gives $(q^{+}, p^{+}) = (h, 1)$ by step 1.1. Thus $H (q^{+}, p^{+}) - H (q, p) = h^{2} / 2 > 0$ , so the original Hamiltonian is not conserved by every step.

Depends on

Used by

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Dependency tree · two levels

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Sources

Ernst Hairer, Geometric Numerical Integration, TU München Lecture 2: Symplectic integrators (standard reference, not scraped)
Nikolaj Nordkvist and Poul G. Hjorth, Classical Mechanics and Symplectic Integration (standard reference, not scraped)